Submanifolds satisfying equations of mean curvature type

In differential geometry, the surfaces with constant mean curvature are the solutions to the isoperimetric problem defined as ``finding those with the smallest area among all compact surfaces with the same volume''. A similar problem for minimal surfaces is ``to characterize those with the smallest area of all surfaces with the same boundary''. In this study, we consider the surfaces that satisfy the singular minimal surface equation, which is an equation of mean curvature type. Such surfaces generalize the minimal surfaces and are critical points of a variational integral called potential $\alpha-$energy. In other words, they minimize the potential $\alpha-$energy and create surface models with the lowest center of gravity. For this reason, they have a great importance in physics and architecture. They also characterize two and higher dimensional analogues of catenary that minimize the potential energy under the influence of gravity. Accordingly, the main problems we deal with in this study are: classification of singular minimal translation hypersurfaces in $(n+1)$-dimensional Euclidean space, classification of singular minimal surfaces with certain semi-symmetric metric (respectively, non-metric) connections and classification of singular minimal surfaces that are minimal with symmetric metric (respectively, non-metric) connections in $3-$dimensional Euclidean and Lorentz--Minkowski spaces. The main purpose of our thesis is to bring new surface samples belonging to this class of surfaces to the literature with the solutions of these problems.